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It is well-known that the law of total probability does not hold in generalin quantum theory. However, the recent arguments on some of the fundamentalassumptions in quantum theory based on the extended Wigner's Friend scenarioshow a need to clarify how the law of total probability should be formulated inquantum theory and under what conditions it still holds. In this work, thedefinition of conditional probability in quantum theory is extended to POVMmeasurements. Rule to assign two-time conditional probability is proposed forincompatible POVM operators, which leads to a more general and preciseformulation of the law of total probability. Sufficient conditions under whichthe law of total probability holds are identified. Applying the theorydeveloped here to analyze several quantum no-go theorems related to theextended Wigner's friend scenario reveals logical loopholes in these no-gotheorems. The loopholes exist as a consequence of taking for granted thevalidity of the law of total probability without verifying the sufficientconditions. Consequently, the contradictions in these no-go theorems onlyreconfirm the invalidity of the law of total probability in quantum theory,rather than invalidating the physical statements that the no-go theoremsattempt to refute.

Law of Total Probability in Quantum Theory and Its Application in Wigner's Friend Scenario